Unit (ring theory)

Unit (ring theory)
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are and (from the German term ).
Less commonly, the term unit is sometimes used to refer to the element of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng.
Examples
The multiplicative identity and its additive inverse are always units. More generally, any root of unity in a ring is a unit: if , then is a multiplicative inverse of .
In a nonzero ring, the element 0 is not a unit, so is not closed under addition.
A nonzero ring in which every nonzero element is a unit (that is) is called a division ring (or a skew-field). A commutative division ring is called a field. For example, the unit group of the field of real numbers is .
Integer ring
In the ring of integers , the only units are and .
In the ring of integers modulo , the units are the congruence classes represented by integers coprime to . They constitute the multiplicative group of integers modulo .
Ring of integers of a number field
In the ring obtained by adjoining the quadratic integer to , one has , so is a unit, and so are its powers, so has infinitely many units.
More generally, for the ring of integers in a number field , Dirichlet's unit theorem states that is isomorphic to the group
where is the (finite, cyclic) group of roots of unity in and , the rank of the unit group, is
where are the number of real embeddings and the number of pairs of complex embeddings of , respectively.
This recovers the example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since .
Polynomials and power series
For a commutative ring , the units of the polynomial ring are the polynomials
such that is a unit in and the remaining coefficients are nilpotent, i.e., satisfy for some N.
In particular, if is a domain (or more generally reduced), then the units of are the units of .
The units of the power series ring are the power series
such that is a unit in .
Matrix rings
The unit group of the ring of matrices over a ring is the group of invertible matrices. For a commutative ring , an element of is invertible if and only if the determinant of is invertible in . In that case, can be given explicitly in terms of the adjugate matrix.
In general
For elements and in a ring , if is invertible, then is invertible with inverse ; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:
See Hua's identity for similar results.
Group of units
A commutative ring is a local ring if is a maximal ideal.
As it turns out, if is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from .
If is a finite field, then is a cyclic group of order .
Every ring homomorphism induces a group homomorphism , since maps units to units. In fact, the formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral group ring construction.
The group scheme is isomorphic to the multiplicative group scheme over any base, so for any commutative ring , the groups and are canonically isomorphic to . Note that the functor (that is) is representable in the sense: for commutative rings (this for instance follows from the aforementioned adjoint relation with the group ring construction). Explicitly this means that there is a natural bijection between the set of the ring homomorphisms and the set of unit elements of (in contrast, represents the additive group , the forgetful functor from the category of commutative rings to the category of abelian groups).
Associatedness
Suppose that is commutative. Elements and of are called if there exists a unit in such that ; then write . In any ring, pairs of additive inverse elements and are associate. For example, 6 and −6 are associate in . In general, is an equivalence relation on .
Associatedness can also be described in terms of the action of on via multiplication: Two elements of are associate if they are in the same -orbit.
In an integral domain, the set of associates of a given nonzero element has the same cardinality as .
The equivalence relation can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring .
See also
S-units
Localization of a ring and a module
Notes
Citations
Sources
1 (number)
Algebraic number theory
Group theory
Ring theory
Algebraic properties of elements